On the Capacity Achieving Covariance Matrix for Frequency Selective MIMO Channels Using the Asymptotic Approach

TL;DR

This paper introduces an asymptotic approximation-based algorithm to efficiently compute the capacity-achieving covariance matrix for frequency selective MIMO channels, reducing computational complexity while maintaining accuracy.

Contribution

It proposes a novel large system approximation for the mutual information, enabling a more efficient optimization of the input covariance matrix in frequency selective MIMO channels.

Findings

The approximation is strictly concave and closely matches the true capacity.

The iterative waterfilling algorithm converges reliably.

Results are consistent with direct maximization even for moderate antenna numbers.

Abstract

In this contribution, an algorithm for evaluating the capacity-achieving input covariance matrices for frequency selective Rayleigh MIMO channels is proposed. In contrast with the flat fading Rayleigh case, no closed-form expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding. In this paper, it is proposed to optimize (w.r.t. the input covariance matrix) a large system approximation of the average mutual information derived by Moustakas and Simon. The validity of this asymptotic approximation is clarified thanks to Gaussian large random matrices methods. It is shown that the approximation is a strictly concave function of the input covariance matrix and that the average mutual information evaluated at the argmax of the approximation is equal to the capacity of the channel up to a O(1/t) term, where t is the number of transmit antennas. An algorithm based on an iterative waterfilling scheme is proposed to maximize the average mutual information approximation, and its convergence studied. Numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information.